If only a pointer to that node is given:
1. Single Circular Linked list: You have to traverse the whole list and come back to that node(so as to link previous and next node of the node to be deleted). So, in that case, O(n).
2. Doubly circular Linked list: Just link the next of previous node with the next node of the node to be deleted, and also link the previous node of the next node with node before the node to be deleted. This will take O(1).
P.S. The comments give the same answer too.